I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

Stability analysis and observer design for discrete-time SEIR epidemic models

This paper applies Micken's discretization method to obtain a discrete-time SEIR epidemic model. The positivity of the model along with the existence and stability of equilibrium points is discussed for the discrete-time case. Afterwards, the design of a state observer for this discrete-time SEIR epidemic model is tackled. The analysis of the model along with the observer design is faced in an implicit way instead of obtaining first an explicit formulation of the system which is the novelty of the presented approach. Moreover, some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality that can be cast in the form of a LMI. The feasibility of the matrix inequality is proved, while some simulation examples show the operation and usefulness of the observer.

On the Stability and Equilibrium Points of Multistaged SI (n) R Epidemic Models

This paper relies on the concept of next generation matrix defined ad hoc for a new proposed extended SEIR model referred to as SI(n)R-model to study its stability. The model includes n successive stages of infectious subpopulations, each one acting at the exposed subpopulation of the next infectious stage in a cascade global disposal where each infectious population acts as the exposed subpopulation of the next infectious stage. The model also has internal delays which characterize the time intervals of the coupling of the susceptible dynamics with the infectious populations of the various cascade infectious stages. Since the susceptible subpopulation is common, and then unique, to all the infectious stages, its coupled dynamic action on each of those stages is modeled with an increasing delay as the infectious stage index increases from 1 to n. The physical interpretation of the model is that the dynamics of the disease exhibits different stages in which the infectivity and the mortality rates vary as the individual numbers go through the process of recovery, each stage with a characteristic average time.

Nonparametric Bayesian sparse factor models with application to gene expression modeling

A nonparametric Bayesian extension of Factor Analysis (FA) is proposed where observed data $\mathbf{Y}$ is modeled as a linear superposition, $\mathbf{G}$, of a potentially infinite number of hidden factors, $\mathbf{X}$. The Indian Buffet Process (IBP) is used as a prior on $\mathbf{G}$ to incorporate sparsity and to allow the number of latent features to be inferred. The model's utility for modeling gene expression data is investigated using randomly generated data sets based on a known sparse connectivity matrix for E. Coli, and on three biological data sets of increasing complexity.

Ultra-structural defects cause low bone matrix stiffness despite high mineralization in osteogenesis imperfecta mice.

Bone is a complex material with a hierarchical multi-scale organization from the molecule to the organ scale. The genetic bone disease, osteogenesis imperfecta, is primarily caused by mutations in the collagen type I genes, resulting in bone fragility. Because the basis of the disease is molecular with ramifications at the whole bone level, it provides a platform for investigating the relationship between structure, composition, and mechanics throughout the hierarchy. Prior studies have individually shown that OI leads to: 1. increased bone mineralization, 2. decreased elastic modulus, and 3. smaller apatite crystal size. However, these have not been studied together and the mechanism for how mineral structure influences tissue mechanics has not been identified. This lack of understanding inhibits the development of more accurate models and therapies. To address this research gap, we used a mouse model of the disease (oim) to measure these outcomes together in order to propose an underlying mechanism for the changes in properties. Our main finding was that despite increased mineralization, oim bones have lower stiffness that may result from the poorly organized mineral matrix with significantly smaller, highly packed and disoriented apatite crystals. Using a composite framework, we interpret the lower oim bone matrix elasticity observed as the result of a change in the aspect ratio of apatite crystals and a disruption of the crystal connectivity.

Models for acoustic propagation through turbofan exhaust flows

Turbomachinery noise radiating into the rearward arc is an important problem. This noise is scattered by the trailing edges of the nacelle and the jet exhaust, and interacts with the shear layers between the external flow, bypass stream and jet, en route to the far field. In the past a range of relevant model problems involving semi-infinite cylinders have been solved. However, one limitation of previous solutions is that they do not allow for the jet nozzle to protrude a finite distance beyond the end of the nacelle (or in certain configurations being buried a finite distance upstream). In this paper we use the matrix Wiener-Hopf technique, which will allow precisely the finite nacelle-jet nozzle separation to be included. The crucial step in our work is to factorise a certain matrix as a product of terms analytic and invertible in the upper/lower halves of the complex plane. The way we do this matrix factorisation is quite different in the buried and protruding nozzle cases. In the buried case our solution method is the so-called pole-removal technique. In the technically more demanding protruding case, however, we must first use Pade approximants to generate a uniformly-valid, meromorphic representation of a certain function, before the same pole-removal method can be applied. Sample results are presented, investigating in particular the effects of exit plane stagger. © 2007 by B Veitch and N Peake.

Effects of Ionizing Radiation on Three-Dimensional Human Vessel Models: Differential Effects According to Radiation Quality and Cellular Development

Little is known about the effects of space radiation on the human body. There are a number of potential chronic and acute effects, and one major target for noncarcinogenic effects is the human vasculature. Cellular stress, inflammatory response, and other radiation effects on endothelial cells may affect vascular function. This study was aimed at understanding the effects of space ionizing radiation on the formation and maintenance of capillary-like blood vessels. We used a 3D human vessel model created with human endothelial cells in a gel matrix to assess the effects of low-LET protons and high-LET iron ions. Iron ions were more damaging and caused significant reduction in the length of intact vessels in both developing and mature vessels at a dose of 80 cGy. Protons had no effect on mature vessels up to a dose of 3.2 Gy but did inhibit vessel formation at 80 cGy. Comparison with gamma radiation showed that photons had even less effect, although, as with protons, developing vessels were more sensitive. Apoptosis assays showed that inhibition of vessel development or deterioration of mature vessels was not due to cell death by apoptosis even in the case of iron ions. These are the first data to show the effects of radiation with varying linear energy transfer on a human vessel model. (C) 2011 In Radiation Research Society

Computer simulation of the damage evolution of fiber-reinforced polypropylene-matrix composites with matrix defects

The damage evolution of fiber-reinforced polypropylene-matrix composites with matrix defects was studied via a Monte Carlo technique combined with a finite element method. A finite element model was constructed to predict the effects of various matrix defect shapes on the stress distributions. The results indicated that a small matrix defect had almost no effect on fiber stress distributions other than interfacial shear stress distributions. Then, a finite element model with a statistical distribution of the fiber strength was constructed to investigate the influences of the spatial distribution and the volume fraction of matrix defects on composite failure. The results showed that it was accurate to use the shear-lag models and Green's function methods to predict the tensile strength of composites even though the axial stresses in the matrix were neglected.

Numerical simulation of mesoscopic mechanical behaviors of gradual multi-fiber-reinforced polymer matrix composites

To simulate the deformation and the fracture of gradual multi-fiber-reinforced matrix composites, a numerical simulation method for the mesoscopic mechanical behaviors was developed on the basis of the finite element and the Monte Carlo methods. The results indicate that the strength of a composite increases if the variability of statistical fiber strengths is decreased.

Iterative Linearized Density Matrix Propagation for Modeling Coherent Energy Transfer in Photosynthetic Light Harvesting

We present results of calculations [1] that employ a new mixed quantum classical iterative density matrix propagation approach (ILDM , or so called Is‐Landmap) [2] to explore the survival of coherence in different photo synthetic models. Our model studies confirm the long lived quantum coherence , while conventional theoretical tools (such as Redfield equation) fail to describe these phenomenon [3,4]. Our ILDM method is a numerical exactly propagation scheme and can be served as a bench mark calculation tools[2]. Result get from ILDM and from other recent methods have been compared and show agreement with each other[4,5]. Long lived coherence plateau has been attribute to the shift of harmonic potential due to the system bath interaction, and the harvesting efficiency is a balance between the coherence and dissipation[1]. We use this approach to investigate the excitation energy transfer dynamics in various light harvesting complex include Fenna‐Matthews‐Olsen light harvesting complex[1] and Cryptophyte Phycocyanin 645 [6]. [1] P.Huo and D.F.Coker ,J. Chem. Phys. 133, 184108 (2010) . [2] E.R. Dunkel, S. Bonella, and D.F. Coker, J. Chem. Phys. 129, 114106 (2008). [3] A. Ishizaki and G.R. Fleming, J. Chem. Phys. 130, 234111 (2009). [4] A. Ishizaki and G.R. Fleming, Proc. Natl. Acad. Sci. 106, 17255 (2009). [5] G. Tao and W.H. Miller, J. Phys. Chem. Lett. 1, 891 (2010). [6] P.Huo and D.F.Coker in preparation

Interaction strengths and net effects in food web models

Understanding how dynamic ecological communities respond to anthropogenic drivers of change such as habitat loss and fragmentation, climate change and the introduction of alien species requires that there is a theoretical framework able to predict community dynamics. At present there is a lack of empirical data that can be used to inform and test predictive models, which means that much of our knowledge regarding the response of ecological communities to perturbations is obtained from theoretical analyses and simulations. This thesis is composed of two strands of research: an empirical experiment conducted to inform the scaling of intraspecific and interspecific interaction strengths in a three species food chain and a series of theoretical analyses on the changes to equilibrium biomass abundances following press perturbations. The empirical experiment is a consequence of the difficulties faced when parameterising the intraspecific interaction strengths in a Lotka-Volterra model. A modification of the dynamic index is used alongside the original dynamic index to estimate intraspecific interactions and interspecific interaction strengths in a three species food. The theoretical analyses focused on the effect of press perturbations to focal species on the equilibrium biomass densities of all species in the community; these perturbations allow for the quantification of a species total net effect. It was found that there is a strong and consistent positive relationship between a species body size and its total net effect for a set of 97 synthetic food webs and also for the Ythan Estuary and Tuesday Lake food webs (empirically described food webs). It is shown that ecological constraints (due to allometric scaling) on the magnitude of entries in the community matrix cause the patterns observed in the inverse community matrix and thus explain the relationship between a species body mass and its total net effect in a community.

Assessment of LD matrix measures for the analysis of biological pathway association.

Complex diseases will have multiple functional sites, and it will be invaluable to understand the cross-locus interaction in terms of linkage disequilibrium (LD) between those sites (epistasis) in addition to the haplotype-LD effects. We investigated the statistical properties of a class of matrix-based statistics to assess this epistasis. These statistical methods include two LD contrast tests (Zaykin et al., 2006) and partial least squares regression (Wang et al., 2008). To estimate Type 1 error rates and power, we simulated multiple two-variant disease models using the SIMLA software package. SIMLA allows for the joint action of up to two disease genes in the simulated data with all possible multiplicative interaction effects between them. Our goal was to detect an interaction between multiple disease-causing variants by means of their linkage disequilibrium (LD) patterns with other markers. We measured the effects of marginal disease effect size, haplotype LD, disease prevalence and minor allele frequency have on cross-locus interaction (epistasis). In the setting of strong allele effects and strong interaction, the correlation between the two disease genes was weak (r=0.2). In a complex system with multiple correlations (both marginal and interaction), it was difficult to determine the source of a significant result. Despite these complications, the partial least squares and modified LD contrast methods maintained adequate power to detect the epistatic effects; however, for many of the analyses we often could not separate interaction from a strong marginal effect. While we did not exhaust the entire parameter space of possible models, we do provide guidance on the effects that population parameters have on cross-locus interaction.

The role of extracellular matrix elasticity and composition in regulating the nucleus pulposus cell phenotype in the intervertebral disc: a narrative review.

Intervertebral disc (IVD) disorders are a major contributor to disability and societal health care costs. Nucleus pulposus (NP) cells of the IVD exhibit changes in both phenotype and morphology with aging-related IVD degeneration that may impact the onset and progression of IVD pathology. Studies have demonstrated that immature NP cell interactions with their extracellular matrix (ECM) may be key regulators of cellular phenotype, metabolism and morphology. The objective of this article is to review our recent experience with studies of NP cell-ECM interactions that reveal how ECM cues can be manipulated to promote an immature NP cell phenotype and morphology. Findings demonstrate the importance of a soft (<700 Pa), laminin-containing ECM in regulating healthy, immature NP cells. Knowledge of NP cell-ECM interactions can be used for development of tissue engineering or cell delivery strategies to treat IVD-related disorders.

In vitro release of nonoxynol-9 from silicone matrix intravaginal rings

The controlled-release characteristics of matrix silicone intravaginal rings loaded with between 100 and 971 mg of nonoxynol-9 have been investigated with a view to developing a ring that may offer a new female-controlled method for the prevention of transmission of sexually transmitted diseases, particularly HIV. Intravaginal rings containing 253, 487 and 971 mg of nonoxynol-9 provided a daily release of 2 mg or more over the 8-day release period, the minimal amount of nonoxynol-9 considered to provide an effective vaginal concentration for the prevention of HIV. Furthermore, the maximum daily release of N9 was about 6 mg, an amount significantly smaller than that observed for other nonoxynol-9 products whose large daily doses may in fact increase the occurrence of HIV by causing epithelial damage to the vaginal tissue. The release mechanism of the liquid nonoxynol-9 from the intravaginal rings has also been investigated and compared to models describing the release of solid drugs from the rings. It has been demonstrated through release studies and surface microscopy that a drug depletion zone is not established in such liquid-loaded intravaginal ring systems, with implications for the release kinetics. (C) 2003 Elsevier B.V. All rights reserved.